1) Is this legal/normal?
It's not common. Whether it's legal is a legal question, not a tax question. We can't answer that here.
2) Should this be reported to me on a tax form?
Who is "me"? Nothing about your wife's income should be reported to you. It should be reported to her. What should be reported is the actual amount that was paid to her, not including the phantom $20 that she never received.
3) Can I claim it as a business expense?
No. If I understand you correctly, the 1099-NEC that your wife gets shows only the $30 that she actually received. In that case the $20 is already deducted from the amount of income that she reports. She can't deduct it again on her tax return.
It sounds like the $20 in the contract is just smoke and mirrors. Your wife is paid $30 and she reports $30 of income. The $20 is not a real expense. It's not clear what someone at the school is trying to accomplish by pretending they are paying her $50. It might be some kind of attempt to cheat on their taxes, but that's not your concern.
